Share This Article:

On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function

Full-Text HTML XML Download Download as PDF (Size:188KB) PP. 35-40
DOI: 10.4236/eng.2013.55A006    4,600 Downloads   6,116 Views   Citations


In 2000, Kostyrko, Salat, and Wilczynski introduced and studied the concept of I-convergence of sequences in metric spaces where I is an ideal. The concept of I-convergence has a wide application in the field of Number Theory, trigonometric series, summability theory, probability theory, optimization and approximation theory. In this article we introduce the double sequence spaces and ,for a modulus function f and study some of the properties of these spaces.

Cite this paper

V. Khan and N. Khan, "On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function," Engineering, Vol. 5 No. 5A, 2013, pp. 35-40. doi: 10.4236/eng.2013.55A006.


[1] H. Fast, “Sur la Convergence Statistique,” Colloqium Mathematicum, Vol. 2, No. 1, 1951, pp. 241-244.
[2] J. A. Fridy, “On Statistical Convergence,” Analysis, Vol. 5, 1985, pp. 301-313.
[3] J. A. Fridy, “Statistical Limit Points,” Proceedings of American Mathematical Society, Vol. 11, 1993, pp. 11871192. doi:10.1090/S0002-9939-1993-1181163-6
[4] P. Kostyrko, T. Salat and W. Wilczynski, “I-Convergence,” Real Analysis Exchange, Vol. 26, No. 2, 1999, pp. 193-200.
[5] T. Salat, B. C. Tripathy and M. Ziman, “On Some Properties of I-Convergence,” Tatra Mountain Mathematical Publications, 2000, pp. 669-686.
[6] K. Demirci, “I-Limit Superior and Limit Inferior,” Mathematical Communications, Vol. 6, 2001, pp. 165-172.
[7] T. J. I. Bromwich, “An Introduction to the Theory of Infinite Series,” MacMillan Co. Ltd., New York, 1965.
[8] M. Basarir and O. Solancan, “On Some Double Sequence Spaces,” Journal of the Indian Academy of Mathematics, Vol. 21, No. 2, 1999, pp. 193-200.
[9] H. Nakano, “Concave Modulars,” Journal of Mathematical Society, Japan, Vol. 5, No. 1, 1953, pp. 29-49. doi:10.2969/jmsj/00510029
[10] W. H. Ruckle, “On Perfect Symmetric BK-Spaces,” Mathematische Annalen, Vol. 175, No. 2, 1968, pp. 121-126. doi:10.1007/BF01418767
[11] W. H. Ruckle, “FK-Spaces in Which the Sequence of Coordinate Vectors is Bounded,” Canadian Journal of Mathematics, Vol. 25, No. 5, 1973, pp. 973-975. doi:10.4153/CJM-1973-102-9
[12] B. Gramsch, “Die Klasse Metrisher Linearer Raume L(φ),” Mathematische Annalen, Vol. 171, 1967, pp. 6178. doi:10.1007/BF01433094
[13] D. J. H. Garling, “On Symmetric Sequence Spaces,” Proceedings of London Mathematical Society, Vol. 16, 1966, pp. 85-106. doi:10.1112/plms/s3-16.1.85
[14] D. J. H. Garling, “Symmetric Bases of Locally Convex Spaces,” Studia Mathematica, Vol. 30, No. 2, 1968, pp. 163-181.
[15] G. Kothe, “Topological Vector Spaces,” Springer, Berlin, 1970.
[16] W. H. Ruckle, “Symmetric Coordinate Spaces and Symmetric Bases,” Canadian Journal of Mathematics, Vol. 19, 1967, pp. 828-838. doi:10.4153/CJM-1967-077-9
[17] V. A. Khan and S. Tabassum, “On Some New Double Sequence Spaces of Invariant Means Defined by Orlicz Function,” Communications, Faculty of Sciences, University of Ankara, Vol. 60, 2011, pp. 11-21.
[18] J. Singer, “Bases in Banach Spaces. 1,” Springer, Berlin, 1970.
[19] M. Sen and S. Roy, “Some I-Convergent Double Classes of Sequences of Fuzzy Numbers Defined by Orlicz Functions,” Thai Journal of Mathematics, Vol. 10, No. 4, 2013, pp. 1-10.
[20] I. J. Maddox, “Some Properties of Paranormed Sequence Spaces,” Journal of the London Mathematical Society, Vol. 1, 1969, pp. 316-322.
[21] J. Connor and J. Kline, “On Statistical Limit Points and the Consistency of Statistical Convergence,” Journal of Mathematical Analysis and Applications, Vol. 197, No. 2, 1996, pp. 392-399. doi:10.1006/jmaa.1996.0027
[22] K. Dems, “On I-Cauchy Sequences,” Real Analysis Exchange, Vol. 30, No. 1, 2005, pp. 123-128.
[23] M. Gurdal, “Some Types Of Convergence,” Doctoral Dissertation, Sleyman Demirel University, Isparta, 2004.
[24] O. T. Jones and J. R. Retherford, “On Similar Bases in Barrelled Spaces,” Proceedings of American Mathematical Society, Vol. 18, 1967, pp. 677-680. doi:10.1090/S0002-9939-1967-0217552-8
[25] P. K. Kamthan and M. Gupta, “Sequence Spaces and Series,” Marcel Dekker Inc., New York, 1981.
[26] I. J. Maddox, “Elements of Functional Analysis,” Cambridge University Press, Cambridge, 1970.
[27] I. J. Maddox, “Sequence Spaces Defined by a Modulus,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 100, 1986, pp. 161-166. doi:10.1017/S0305004100065968
[28] T. Salat, “On Statistically Convergent Sequences of Real Numbers,” Mathematica Slovaca, Vol. 30, 1980, pp. 139150.
[29] A. K. Vakeel and K. Ebadullah, “On Some I-Convergent Sequence Spaces Defined by a Modulus Function,” Theory and Applications of Mathematics and Computer Science, Vol. 1, No. 2, 2011, pp. 22-30.
[30] A. Wilansky, “Functional Analysis,” Blaisdell, New York, 1964.

comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.